Solving matrix inequalities whose unknowns are matrices

This paper provides algorithms for numerical solution of convex matrix inequalities (MIs) in which the variables naturally appear as matrices. This includes, for instance, many systems and control problems. To use these algorithms, no knowledge of linear matrix inequalities (LMIs) is required. However, as tools, they preserve many advantages of the linear matrix inequality framework. Our method has two components: 1) a numerical (partly symbolic) algorithm that solves a large class of matrix optimization problems; 2) a symbolic ??Convexity Checker?? that automatically provides a region which, if convex, guarantees that the solution from (1) is a global optimum on that region.